Hardware Implementation of Large Number-Multiplication by FFT with Modular Arithmetic
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Bibliographic record
Abstract
Modular multiplication (MM) for large integers is the foundation of most public-key cryptosystems, specifically RSA, El-Gamal and the elliptic curve cryptosystems. Thus MM algorithms have been studied widely and extensively. Most of works are based on the well known Montgomery multiplication method (MMM) and its variants, which require multiplication in N. Authors have always avoided the fast Fourier transform (FFT) method believing that it is impractical for present system sizes despite its smaller complexity order. In this paper, the authors presented the design and hardware implementation of a FFT-based algorithm using modular arithmetic to efficiently compute very large number multiplications. The algorithm has been implemented in CASM, an intermediate level HDL developed in the laboratory. The target architecture is a FPGA. The algorithm is scalable and can easily be mapped to any operand size. Results show that such algorithm implementation starts to be useful for 4096-bit operands and beyond.
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Full frame distilled prediction
Teacher imitationNot calibrated prevalence, not ground truth. Human validation pending. Learned from the 10,348 direct Codex labels and 10,348 direct Gemma labels. Candidate is the union of thresholded teacher heads; consensus is their intersection. These outputs are machine_predicted_unvalidated and are not human labels or direct frontier model labels.
Codex and Gemma teacher scores by category
| Category | Codex | Gemma |
|---|---|---|
| Metaresearch | 0.000 | 0.000 |
| Meta-epidemiology (narrow) | 0.000 | 0.000 |
| Meta-epidemiology (broad) | 0.000 | 0.000 |
| Bibliometrics | 0.000 | 0.000 |
| Science and technology studies | 0.000 | 0.000 |
| Scholarly communication | 0.000 | 0.000 |
| Open science | 0.000 | 0.000 |
| Research integrity | 0.000 | 0.000 |
| Insufficient payload (model declined to judge) | 0.000 | 0.000 |
Machine scores (provisional)
The two teacher heads of the student model, read on this work. A score orders the frame for review; it never asserts a category, and the validation status ships verbatim with every row.
Baseline scores from an immature model (maturity gate not passed, 7 training rounds). Scores rank; they never assert a category.
score_only:v0-immature-baseline · verbatim from the scoring run: score_only means the number may rank works, and no category label ships from it